Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments Hanen Amor a , Jean-Jacques Marigo b , Corrado Maurini b, a LPMTM (CNRS-UPR 9001) and LAGA (CNRS-UMR 7539), Institut Galile´e, Univ Paris-Nord, 99 avenue Jean-Baptiste Cle´ment, 93430 Villetaneuse, France b Institut Jean Le Rond d’Alembert, UPMC Univ Paris 06 and CNRS (UMR 7190), 4 place Jussieu, 75252 Paris Cedex 05, France article info Article history: Received 10 January 2009 Received in revised form 23 April 2009 Accepted 24 April 2009 Keywords: Fracture Energy methods Free-discontinuity problems Contact mechanics Finite elements abstract This paper presents a modified regularized formulation of the Ambrosio–Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8),1319–1342]. We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches. & 2009 Elsevier Ltd. All rights reserved. 1. Introduction Computational approaches for simulating fracture of solids at the macroscopic scale are commonly classified into two categories (see e.g. de Borst et al., 2004): (i) discrete crack models with an explicit geometric modeling of cracks as surfaces of discontinuities and (ii) smeared crack models, approximating cracks with continuum fields having high gradients localized in thin bands. The implementation of discrete crack approaches requires specific techniques to introduce discontinuous fields in the numerical model. The classical method consists in changing the mesh geometry by introducing new boundaries as the crack propagates together with adaptive remeshing (Ingraffea and Saouma, 1984). Efficient alternatives are the extended finite element methods (Moe ¨s et al., 1999), which enrich the finite element shape functions with discontinuous fields on the basis of a partition of unity concept (Babuska and Melenk, 1997), and interelement crack methods (Xu and Needleman, 1994; Camacho and Ortiz, 1996), which constrain cracks to propagate along the element interfaces. Smeared crack (or continuum) approaches, include damage models (see e.g. Jirasek, 1998; Pijaudier-Cabot and Bazant, 1987; Lorentz and Andrieux, 1999) and diffuse interface (or phase-field) models (Aranson et al., 2000; Hakim and Karma, 2009; Marconi and Jagla, 2005). In the comparative study of Song et al. (2008), these approaches are synthetically classified as element deletion methods. They are based on the use of phenomenological constitutive laws with strain-softening. It is Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jmps Journal of the Mechanics and Physics of Solids ARTICLE IN PRESS 0022-5096/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2009.04.011 Corresponding author. Tel.: +33144 27 87 19; fax: +33144 27 52 59. E-mail address: [email protected] (C. Maurini). Journal of the Mechanics and Physics of Solids 57 (2009) 1209–1229
Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments
May 23, 2023
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